Amyloid dataset in Stat2Data package
Abeta: Amyloid-\(\beta\) in posterior cingulate cortex (pmol/g tissue)Group:
mAD = Alzheimer’s diseaseMCI = mild cognitive impairmentNCI = no cognitive impairment| Var1 | Freq | Prob |
|---|---|---|
| mAD | 17 | 0.298 |
| MCI | 21 | 0.368 |
| NCI | 19 | 0.333 |
mAD, MCI, NCI
mAD is referenceMCI vs mADNCI vs mADGroup\[ln\left(\frac{\hat{p}_{MCI}}{\hat{p}_{mAD}}\right) = b_{0,2.1} + b_{1,2.1} Abeta\]
\[ln\left(\frac{\hat{p}_{NCI}}{\hat{p}_{mAD}}\right) = b_{0,3.1} + b_{1,3.1} Abeta\]
# weights: 9 (4 variable)
initial value 62.620900
final value 57.318323
converged
Call:
multinom(formula = Group ~ Abeta, data = Amyloid)
Coefficients:
(Intercept) Abeta
MCI 1.319761 -0.002092564
NCI 1.231750 -0.002128210
Std. Errors:
(Intercept) Abeta
MCI 0.5583894 0.0008282646
NCI 0.5666824 0.0008558914
Residual Deviance: 114.6366
AIC: 122.6366
| (Intercept) | Abeta | |
|---|---|---|
| MCI | 1.320 | -0.002 |
| NCI | 1.232 | -0.002 |
| (Intercept) | Abeta | |
|---|---|---|
| MCI | 0.018 | 0.012 |
| NCI | 0.030 | 0.013 |
| 2.5 %.MCI | 97.5 %.MCI | 2.5 %.NCI | 97.5 %.NCI | |
|---|---|---|---|---|
| (Intercept) | 0.225 | 2.414 | 0.121 | 2.342 |
| Abeta | -0.004 | 0.000 | -0.004 | 0.000 |
\[ln\left(\frac{\hat{p}_{MCI}}{\hat{p}_{mAD}}\right) = b_{0,2.1} + b_{1,2.1} Abeta = 1.32 + (-0.00209) Abeta\]
\[ln\left(\frac{\hat{p}_{NCI}}{\hat{p}_{mAD}}\right) = b_{0,3.1} + b_{1,3.1} Abeta = 1.232 + (-0.00213) Abeta\]
Abeta has a certain effect on the probability of having mild impairment vs Alzheimer’s (\(b_{1,2.1}\))Abeta has a different effect on the probability of having no impairment vs Alzheimer’s (\(b_{1,3.1}\))MCI vs mAD = \(e^{b_{0,2.1}} = e^{1.32} = 3.743\)
Abeta = 0: Odds of MCI is 3.743 times higher than odds of mAD
NCI vs mAD = \(e^{b_{0,3.1}} = e^{1.232} = 3.427\)
Abeta = 0: Odds of NCI is 3.427 times higher than odds of mAD
Abeta
MCI vs mAD = \(e^{b_{1,2.1}} = e^{-0.0020926} = 0.99791\)
Abeta means lower odds of MCI (relative to mAD)
NCI vs mAD = \(e^{b_{1,3.1}} = e^{-0.0021282} = 0.99787\)
Abeta means lower odds of NCI (relative to mAD)
Warning
Warning
mAD, MCI, NCI
mAD then MCI then NCImAD vs all highermAD and MCI vs all higherGroup\[ln\left(\frac{\hat{p}_{mAD}}{\hat{p}_{MCI} + \hat{p}_{NCI}}\right) = b_{0,1} + -b_{1} Abeta\]
\[ln\left(\frac{\hat{p}_{mAD} + \hat{p}_{MCI}}{\hat{p}_{NCI}}\right) = b_{0,12} + -b_{1} Abeta\]
Call:
polr(formula = Group ~ Abeta, data = Amyloid, Hess = TRUE)
Coefficients:
Value Std. Error t value
Abeta -0.001671 0.0006333 -2.639
Intercepts:
Value Std. Error t value
mAD|MCI -1.6689 0.4323 -3.8602
MCI|NCI 0.0729 0.3618 0.2014
Residual Deviance: 116.6483
AIC: 122.6483
| Value | Std. Error | t value | p value | |
|---|---|---|---|---|
| Abeta | -0.002 | 0.001 | -2.639 | 0.008 |
| mAD|MCI | -1.669 | 0.432 | -3.860 | 0.000 |
| MCI|NCI | 0.073 | 0.362 | 0.201 | 0.840 |
2.5 % 97.5 %
-0.002961670 -0.000509697
Warning
\[ln\left(\frac{\hat{p}_{mAD}}{\hat{p}_{MCI} + \hat{p}_{NCI}}\right) = b_{0,1} + -b_{1} Abeta = 1.669 + (0.002) Abeta\]
\[ln\left(\frac{\hat{p}_{mAD} + \hat{p}_{MCI}}{\hat{p}_{NCI}}\right) = b_{0,12} + -b_{1} Abeta = -0.073 + (0.002) Abeta\]
Abeta has a certain effect on the probability of having Alzheimer’s vs (mild or no cognitive impairment) (\(b_1\))Abeta has the same effect on the probability of having (Alzheimer’s or mild cognitive impairment) vs no cognitive impairment (\(b_1\))mAD vs (MCI and NCI) = \(e^{b_{0,1}} = e^{-1.669} = 0.188\)
Abeta = 0: Odds of mAD is 0.188 times odds of MCI and NCI
mAD and MCI) vs NCI = \(e^{b_{0,12}} = e^{0.073} = 1.076\)
Abeta = 0: Odds of mAD and MCI is 1.076 times odds of NCI
Abeta
Abeta: \(e^{-b_{1}} = e^{0.002} = 1.002\)
Abeta means
mAD relative to (MCI and NCI)
mAD and MCI) relative to NCI
Warning
mAD to MCI or from MCI to NCI
mAD vs all highermAD and MCI vs all higher
| term | estimate | estimate |
|---|---|---|
| Abeta | 0.002 | 0.001 |